Sep 10, 1993 - If A denotes a classical gauge field, or connection, a Wilson loop is simply ...... coefficients of the Alexander polynomial to all of M. Some recent ...
Strings, Loops, Knots and Gauge Fields John C. Baez Department of Mathematics University of California Riverside CA 92521
arXiv:hep-th/9309067v1 10 Sep 1993
September 10, 1993 to appear in Knots and Quantum Gravity, ed. J. Baez, Oxford U. Press
Abstract The loop representation of quantum gravity has many formal resemblances to a background-free string theory. In fact, its origins lie in attempts to treat the string theory of hadrons as an approximation to QCD, in which the strings represent flux tubes of the gauge field. A heuristic path-integral approach indicates a duality between background-free string theories and generally covariant gauge theories, with the loop transform relating the two. We review progress towards making this duality rigorous in three examples: 2d Yang-Mills theory (which, while not generally covariant, has symmetry under all area-preserving transformations), 3d quantum gravity, and 4d quantum gravity. SU (N ) YangMills theory in 2 dimensions has been given a string-theoretic interpretation in the large-N limit by Gross, Taylor, Minahan and Polychronakos, but here we provide an exact string-theoretic interpretation of the theory on R × S 1 for finite N . The string-theoretic interpretation of quantum gravity in 3 dimensions gives rise to conjectures about integrals on the moduli space of flat connections, while in 4 dimensions there may be connections to the theory of 2-tangles.
1
Introduction
The notion of a deep relationship between string theories and gauge theories is far from new. String theory first arose as a model of hadron interactions. Unfortunately this theory had a number of undesirable features; in particular, it predicted massless spin-2 particles. It was soon supplanted by quantum chromodynamics (QCD), which models the strong force by an SU(3) Yang-Mills field. However, string models continued to be popular as an approximation of the confining phase of QCD. Two quarks in a meson, for example, can be thought of as connected by a string-like flux tube in which the gauge field is concentrated, while an excitation of the gauge field alone can be thought of as a looped flux tube. This is essentially a modern reincarnation of Faraday’s notion of “field lines,” but it can be formalized using the notion of Wilson loops. If A denotes a classical gauge field, or connection, a Wilson loop is simply the 1
trace of the holonomy of A around a loop γ in space, typically written in terms of a path-ordered exponential H A tr P e γ . If instead A denotes a quantized gauge field, the Wilson loop may be reinterpreted as an operator on the Hilbert space of states, and applying this operator to the vacuum state one obtains a state in which the Yang-Mills analog of the electric field flows around the loop γ. In the late 1970’s, Makeenko and Migdal, Nambu, Polyakov, and others [37, 41] attempted to derive equations of string dynamics as an approximation to the Yang-Mills equation, using Wilson loops. More recently, D. Gross and others [24, 25, 34, 35, 36] have been able to exactly reformulate Yang-Mills theory in 2-dimensional spacetime as a string theory by writing an asymptotic series for the vacuum expectation values of Wilson loops as a sum over maps from surfaces (the string worldsheet) to spacetime. This development raises the hope that other gauge theories might also be isomorphic to string theories. For example, recent work by Witten [50] and Periwal [40] suggests that Chern-Simons theory in 3 dimensions is also equivalent to a string theory. String theory eventually became popular as a theory of everything because the massless spin-2 particles it predicted could be interpreted as the gravitons one obtains by quantizing the spacetime metric perturbatively about a fixed “background” metric. Since string theory appears to avoid the renormalization problems in perturbative quantum gravity, it is a strong candidate for a theory unifying gravity with the other forces. However, while classical general relativity is an elegant geometrical theory relying on no background structure for its formulation, it has proved difficult to describe string theory along these lines. Typically one begins with a fixed background structure and writes down a string field theory in terms of this; only afterwards can one investigate its background independence [52]. The clarity of a manifestly background-free approach to string theory would be highly desirable. On the other hand, attempts to formulate Yang-Mills theory in terms of Wilson loops eventually led to a full-fledged “loop representation” of gauge theories, thanks to the work of Gambini, Trias [20], and others. After Ashtekar [1] formulated quantum gravity as a sort of gauge theory using the “new variables,” Rovelli and Smolin [44] were able to use the loop representation to study quantum gravity nonperturbatively in a manifestly background-free formalism. While superficially quite different from modern string theory, this approach to quantum gravity has many points of similarity, thanks to its common origin. In particular, it uses the device of Wilson loops to construct a space of states consisting of “multiloop invariants,” which assign an amplitude to any collection of loops in space. The resemblance of these states to wavefunctions of a string field theory is striking. It is natural, therefore, to ask whether the loop representation of quantum gravity might be a string theory in disguise - or vice versa. The present paper does not attempt a definitive answer to this question. Rather, we begin by describing a general framework relating gauge theories and string theories, 2
and then consider a variety of examples. Our treatment of examples is also meant to serve as a review of Yang-Mills theory in 2 dimensions and quantum gravity in 3 and 4 dimensions. In Section 2 we describe how the loop representation of a generally covariant gauge theories is related to a background-free closed string field theory. We take a very naive approach to strings, thinking of them simply as maps from a surface into spacetime, and disregarding any conformal structure or fields propagating on the surface. We base our treatment on the path integral formalism, and in order to simplify the presentation we make a number of over-optimistic assumptions concerning measures on infinite-dimensional spaces such as the space A/G of connections modulo gauge transformations. In Section 3 we consider Yang-Mills theory in 2 dimensions as an example. In fact, this theory is not generally covariant, but it has an infinite-dimensional subgroup of the diffeomorphism group as symmetries, the group of all area-preserving transformations. Rather than the path-integral approach we use canonical quantization, which is easier to make rigorous. Gross, Taylor, Minahan, and Polychronakos [24, 25, 34, 35] have already given 2-dimensional SU(N) Yang-Mills theory a string-theoretic interpretation in the large N limit. Our treatment is mostly a review of their work, but we find it to be little extra effort, and rather enlightening, to give the theory a precise string-theoretic interpretation for finite N. In Section 4 we consider quantum gravity in 3 dimensions. We review the loop representation of this theory and raise some questions about integrals over the moduli space of flat connections on a Riemann surface whose resolution would be desirable for developing a string-theoretic picture of the theory. We also briefly discuss ChernSimons theory in 3 dimensions. These examples have finite-dimensional reduced configuration spaces, so there are no analytical difficulties with measures on infinite-dimensional spaces, at least in canonical quantization. In Section 5, however, we consider quantum gravity in 4 dimensions. Here the classical configuration space is infinite-dimensional and issues of analysis become more important. We review recent work by Ashtekar, Isham, Lewandowski and the author [3, 4, 10] on diffeomorphism-invariant generalized measures on A/G and their relation to multiloop invariants and knot theory. We also note how a string-theoretic interpretation of the theory leads naturally to the study of 2-tangles. Acknowledgements. I would like to thank Abhay Ashtekar, Scott Axelrod, Scott Carter, Paolo Cotta-Ramusino, Louis Crane, Jacob Hirbawi, Jerzy Lewandowski, Renate Loll, Maurizio Martellini, Jorge Pullin, Holger Nielsen, and Lee Smolin for useful discussions. Wati Taylor deserves special thanks for explaining his work on Yang-Mills theory to me. Also, I would like to collectively thank the Center for Gravitational Physics and Geometry for inviting me to speak on this subject.
3
2
String Field/Gauge Field Duality
In this section we sketch a relationship between string field theories and gauge theories. We begin with a nonperturbative Lagrangian description of background-free closed string field theories. From this we derive a Hamiltonian description, which turns out to be mathematically isomorphic to the loop representation of a generally covariant gauge theory. We emphasize that while our discussion here is rigorous, it is schematic, in the sense that some of our assumptions are not likely to hold precisely as stated in the most interesting examples. In particular, by “measure” in this section we will always mean a positive regular Borel measure, but in fact one should work with a more general version of this concept. We discuss these analytical issues more carefully in Section 5. Consider a theory of strings propagating on a spacetime M that is diffeomorphic to R×X, with X a manifold we call “space.” We do not assume a canonical identification of M with R × X, or any other background structure (metric, etc.) on spacetime. We take the classical configuration space of the string theory to be the space M of multiloops in X: [ M= Mn n≥0
with Mn = Maps(nS 1 , X). Here nS 1 denotes the disjoint union of n copies of S 1 , and we write “Maps” to denote the set of maps satisfying some regularity conditions (continuity, smoothness, etc.) to be specified. Let Dγ denote a measure on M and let Fun(M) denote some space of square-integrable functions on M. We assume that Fun(M) and the measure Dγ are invariant both under diffeomorphisms of space and reparametrizations of the strings. That is, both the identity component of the diffeomorphism group of X and the orientation-preserving diffeomorphisms of nS 1 act on M, and we wish Fun(M) and Dγ to be preserved by these actions. Introduce on Fun(M) the “kinematical inner product,” which is just the L2 inner product Z hψ, φikin = ψ(γ)φ(γ) Dγ. M
We assume for convenience that this really is an inner product, i.e. it is nondegenerate. Define the “kinematical state space” Hkin to be the Hilbert space completion of Fun(M) in the norm associated to this inner product. Following ideas from canonical quantum gravity, we do not expect Hkin to be the true space of physical states. In the space of physical states, any two states differing by a diffeomorphism of spacetime are identified. The physical state space thus depends on the dynamics of the theory. Taking a Lagrangian approach, dynamics may be described using in terms of path integrals as follows. Fix a time T > 0. Let P denote
4
the set of “histories,” that is, maps f : Σ → [0, T ] × X, where Σ is a compact oriented 2-manifold with boundary, such that f (Σ) ∩ ∂([0, T ] × X) = f (∂Σ). ′
Given γ, γ ∈ M, we say that f ∈ P is a history from γ to γ ′ if f : Σ → [0, T ] × X and the boundary of Σ is a disjoint union of circles nS 1 ∪ mS 1 , with f |nS 1 = γ,
f |mS 1 = γ ′ .
We fix a measure, or “path integral,” on P(γ, γ ′ ). Following tradition, we write this as eiS(f ) Df , with S(f ) denoting the action of f , but eiS(f ) and Df only appear in the combination eiS(f ) Df . Since we are interested in generally covariant theories, this path integral is assumed to have some invariance properties, which we note below as they are needed. Using the standard recipe in topological quantum field theory, we define the “physical inner product” on Hkin by hψ, φiphys =
Z
M
Z
M
Z
P(γ,γ ′ )
ψ(γ)φ(γ ′ ) eiS(f ) Df DγDγ ′
assuming optimistically that this integral is well-defined. We do not actually assume this is an inner product in the standard sense, for while we assume hψ, ψi ≥ 0 for all ψ ∈ Hkin , we do not assume positive definiteness. The general covariance of the theory should imply that this inner product is independent of the choice of time T > 0, so we assume this as well. Define the space of norm-zero states I ⊆ Hkin by I = {ψ| hψ, ψiphys = 0} = {ψ| hψ, φiphys = 0 for all φ ∈ Hkin } and define the “physical state space” Hphys to be the Hilbert space completion of Hkin /I in the norm associated to the physical inner product. In general I is nonempty, because if g ∈ Diff 0 (X) is a diffeomorphism in the connected component of the identity, we can find a path of diffeomorphisms gt ∈ Diff 0 (M) with g0 = g and gT equal to the identity, and defining ge ∈ Diff([0, T ] × X) by we have hψ, φiphys = = = =
Z
M
Z
M
Z
M
Z
M
Z
M
Z
M
Z
M
Z
M
ge(t, x) = (t, gt (x)),
Z
P(γ,γ ′ )
Z
P(γ,γ ′ )
Z
P(γ,γ ′ )
Z
P(γ,γ ′ )
ψ(γ)φ(γ ′ )eiS(f ) Df DγDγ ′ gψ(gγ)φ(γ ′) eiS(f ) Df DγDγ ′ gψ(gγ)φ(γ ′) eiS(˜gf ) D(gef )D(gγ)Dγ ′ gψ(γ)φ(γ ′ ) eiS(f ) Df DγDγ ′
= hgψ, φiphys 5
for any ψ, φ. Here we are assuming eiS(˜gf ) D(gef ) = eiS(f ) Df,
which is one of the expected invariance properties of the path integral. It follows that I includes the space J, the closure of the span of all vectors of the form ψ − gψ. We can therefore define the (spatially) “diffeomorphism-invariant state space” Hdif f by Hdif f = Hkin /J and obtain Hphys as a Hilbert space completion of Hdif f /K, where K is the image of I in Hdif f . To summarize, we obtain the physical state space from the kinematical state space by taking two quotients: Hkin → Hkin /J = Hdif f Hdif f → Hdif f /K ֒→ Hphys . As usual in canonical quantum gravity and topological quantum field theory, there is no Hamiltonian; instead, all the information about dynamics is contained in the physical inner product. The reason, of course, is that the path integral, which in traditional quantum field theory described time evolution, now describes the physical inner product. The quotient map Hdif f → Hphys , or equivalently its kernel K, plays the role of a “Hamiltonian constraint.” The quotient map Hkin → Hdif f , or equivalently its kernel J, plays the role of the “diffeomorphism constraint,” which is independent of the dynamics. (Strictly speaking, we should call K the “dynamical constraint,” as we shall see that it expresses constraints on the initial data other than those usually called the Hamiltonian constraint, such as the “Mandelstam constraints” arising in gauge theory.) It is common in canonical quantum gravity to proceed in a slightly different manner than we have done here, using subspaces at certain points where we use quotient spaces [44, 45]. For example, Hdif f may be defined as the subspace of Hkin consisting of states invariant under the action of Diff 0 (X), and Hphys then defined as the kernel of certain operators, the Hamiltonian constraints. The method of working solely with quotient spaces, has, however, been studied by Ashtekar [2]. The choice between these different approaches will in the end be dictated by the desire for convenience and/or rigor. As a heuristic guiding principle, however, it is worth noting that the subspace and quotient space approaches are essentially equivalent if we assume that the subspace I is closed in the norm topology on Hkin . Relative to the kinematical inner product, we can identify Hdif f with the orthogonal complement J⊥ , and similarly identify Hphys with I⊥ . From this point of view we have Hphys ⊆ Hdif f ⊆ Hkin . Moreover, ψ ∈ Hdif f if and only if ψ is invariant under the action of Diff 0 (X) on Hkin . To see this, first note that if gψ = ψ for all g ∈ Diff 0 (X), then for all φ ∈ Hkin we have hψ, gφ − φi = hg −1ψ − ψ, φi = 0 6
Hψ(
@ @ @ R @
)
=
aψ(
@ @ @ R @
) + bψ(
@ @
@ R @
-
) + cψ(
) -
Figure 1: Two-string interaction in 3-dimensional space so ψ ∈ J⊥ . Conversely, if ψ ∈ J⊥ , hψ, gψ − ψi = 0 so hψ, ψi = hψ, gψi, and since g acts unitarily on Hkin the Cauchy-Schwarz inequality implies gψ = ψ. The approach using subspaces is the one with the clearest connection to knot theory. An element ψ ∈ Hkin is function on the space of multiloops. If ψ is invariant under the action of Diff 0 (X), we call ψ a “multiloop invariant.” In particular, ψ defines an ambient isotopy invariant of links in X when we restrict it to links (which are nothing but multiloops that happen to be embeddings). We see therefore that in this situation the physical states define link invariants. As a suggestive example, take X = S 3 , and take as the Hamiltonian constraint an operator H on Hdif f that has the property described in Figure 1. Here a, b, c ∈ C are arbitrary. This Hamiltonian constraint represents the simplest sort of diffeomorphism-invariant two-string interaction in 3-dimensional space. Defining the physical space Hphys to be the kernel of H, it follows that any ψ ∈ Hphys gives a link invariant that is just a multiple of the HOMFLY invariant [19]. For appropriate values of the parameters a, b, c, we expect this sort of Hamiltonian constraint to occur in a generally covariant gauge theory on 4-dimensional spacetime known as BF theory, with gauge group SU(N) [27]. A similar construction working with unoriented framed multiloops gives rise to the Kauffman polynomial, which is associated with BF theory with gauge group SO(N) [28]. We see here in its barest form the path from string-theoretic considerations to link invariants and then to gauge theory. In what follows, we start from the other end, and consider a generally covariant gauge theory on M. Thus we fix a Lie group G and a principal G-bundle P → M. Fixing an identification M ∼ = R×X, the classical configuration space is the space A of connections on P |{0}×X . (The physical Hilbert space of the quantum theory, it should be emphasized, is supposed to be independent of this identification M ∼ = R × X.) Given a loop γ: S 1 → X and a connection A ∈ A, let T (γ, A) be the corresponding Wilson loop, that is, the trace of the holonomy of A around γ in a fixed finitedimensional representation of G: H
T (γ, A) = tr P e
γ
A
.
The group G of gauge transformations acts on A. Fix a G-invariant measure DA on A and let Fun(A/G) denote a space of gauge-invariant functions on A containing 7
the algebra of functions generated by Wilson loops. We may alternatively think of Fun(A/G) as a space of functions on A/G and DA as a measure on A/G. Assume that DA is invariant under the action of Diff 0 (M) on A/G, and define the kinematical state space Hkin to be be the Hilbert space completion of Fun(A/G) in the norm associated to the kinematical inner product hψ, φikin =
Z
A/G
ψ(A)φ(A) DA.
The relation of this kinematical state space and that described above for a string field theory is given by the loop transform. Given any multiloop (γ1 , . . . , γn ) ∈ Mn , define the loop transform ψˆ of ψ ∈ Fun(A/G) by ˆ 1 , . . . , γn ) = ψ(γ
Z
A/G
ψ(A)T (γ1 , A) · · · T (γn , A) DA.
Take Fun(M) to be the space of functions in the range of the loop transform. Let us assume, purely for simplicity of exposition, that the loop transform is one-to-one. Then we may identify Hkin with Fun(M) just as in the string field theory case. The process of passing from the kinematical state space to the diffeomorphisminvariant state space and then the physical state space has already been treated for a number of generally covariant gauge theories, most notably quantum gravity [1, 43, 44]. In order to emphasize the resemblance to the string field case, we will use a path integral approach. Fix a time T > 0. Given A, A′ ∈ A, let P(A, A′ ) denote the space of connections on P |[0,T ]×X which restrict to A on {0} × X and to A′ on {T } × X. We assume the existence of a measure on P(A, A′ ) which we write as eiS(a) Da, using a to denote a connection on P |[0,T ]×X . Again, this generalized measure has some invariance properties corresponding to the general covariance of the gauge theory. Define the “physical” inner product on Hkin by hψ, φiphys =
Z Z Z A
A
P(A,A′ )
ψ(A)φ(A′ ) eiS(a) DaDADA′
again assuming that this integral is well-defined and that hψ, ψi ≥ 0 for all ψ. This inner product should be independent of the choice of time T > 0. Letting I ⊆ Hkin denote the space of norm-zero states, the physical state space Hphys of the gauge theory is Hkin /I. As before, we can use the general covariance of the theory to show that I contains the closed span J of all vectors of the form ψ − gψ. Letting Hdif f = Hkin /J, and letting K be the image of I in Hdif f , we again see that the physical state space is obtained by applying first the diffeomorphism constraint Hkin → Hkin /J = Hdif f and then the Hamiltonian constraint Hdif f → Hdif f /K ֒→ Hphys. 8
In summary, we see that the Hilbert spaces for generally covariant string theories and generally covariant gauge theories have a similar form, with the loop transform relating the gauge theory picture to the string theory picture. The key point, again, is that a state ψ in Hkin can either be regarded as a wavefunction on the classical configuration space A for gauge fields, with ψ(A) being the amplitude of a specified connection A, or as a wavefunction on the classical configuration space M ˆ 1 , · · · , γn ) being the amplitude of a specified n-tuple of strings for strings, with ψ(γ 1 γ1 , . . . , γn : S → X to be present. The loop transform depends on the nonlinear “duality” between connections and loops, A/G × M → C (A, (γ1 , . . . , γn )) 7→ T (A, γ) · · · T (A, γn ) which is why we speak of string field/gauge field duality rather than an isomorphism between string fields and gauge fields. At this point it is natural to ask what is the difference, apart from words, between the loop representation of a generally covariant gauge theory and the sort of purely topological string field theory we have been considering. From the Hamiltonian viewpoint (that is, in terms of the spaces Hkin , Hdif f , and Hphys ) the difference is not so great. The Lagrangian for a gauge theory, on the other hand, is quite a different object than that of a string field theory. Note that nothing we have done allows the direct construction of a string field Lagrangian from a gauge field Lagrangian or vice versa. In the following sections we will consider some examples: Yang-Mills theory in 2 dimensions, quantum gravity in 3 dimensions, and quantum gravity in 4 dimensions. In no case is a string field action S(f ) known that corresponds to the gauge theory in question! However, in 2d Yang-Mills theory a working substitute for the string field path integral is known: a discrete sum over certain equivalence classes of maps f : Σ → M. This is, in fact, a promising alternative to dealing with measures on the space P of string histories. In 4 dimensional quantum gravity, such an approach might involve a sum over “2-tangles,” that is, ambient isotopy classes of embeddings f : Σ → [0, T ] × X.
3
Yang-Mills Theory in 2 Dimensions
We begin with an example in which most of the details have been worked out. YangMills theory is not a generally covariant theory since it relies for its formulation on a fixed Riemannian or Lorentzian metric on the spacetime manifold M. We fix a connected compact Lie group G and a principal G-bundle P → M. Classically the gauge fields in question are connections A on P , and the Yang-Mills action is given by Z 1 S(A) = − tr(F ∧ ⋆F ) 2 M
9
where F is the curvature of A and tr is the trace in a fixed faithful unitary representation of G and hence its Lie algebra g. Extremizing this action we obtain the classical equations of motion, the Yang-Mills equation dA ⋆ F = 0, where dA is the exterior covariant derivative. The action S(A) is gauge-invariant so it can be regarded as a function on the space of connections on M modulo gauge transformations. The group Diff(M) acts on this space, but the action is not diffeomorphism-invariant. However, if M is 2-dimensional one may write F = f ⊗ ω where ω is the volume form on M and f is a section of P ×Ad g, and then Z 1 S(A) = − tr(f 2 ) ω. 2 M It follows that the action S(A) is invariant under the subgroup of diffeomorphisms preserving the volume form ω. So upon quantization one expects to - and does - obtain something analogous to a topological quantum field theory, but in which diffeomorphism-invariance is replaced by invariance under this subgroup. Strictly speaking, then, many of the results of the previous section not apply. In particular, this theory one has an honest Hamiltonian, rather than a Hamiltonian constraint. Still, it illustrates some interesting aspects of gauge field/string field duality. The Riemannian case of 2d Yang-Mills theory has been extensively investigated. An equation for the vacuum expectation values of Wilson loops for the theory on Euclidean R2 was found by Migdal [33], and these expectation values were explicitly calculated by Kazakov [29]. These calculations were made rigorous using stochastic differential equation techniques by L. Gross, King and Sengupta [26], as well as Driver [16]. The classical Yang-Mills equations on Riemann surfaces were extensively investigated by Atiyah and Bott [8], and the quantum theory on Riemann surfaces has been studied by Rusakov [46], Fine [17] and Witten [48]. In particular, Witten has shown that the quantization of 2d Yang-Mills theory gives a mathematical structure very close to that of a topological quantum field theory, with a Hilbert space Z(S 1 ∪ · · · ∪ S 1 ) associated to each compact 1-manifold S 1 ∪ · · · ∪ S 1 , and a vector Z(M, α) ∈ Z(∂M) for each compact oriented 2-manifold M with boundary having R total area α = M ω. Let us briefly review some of this work while adapting it to Yang-Mills theory on R × S 1 with the Lorentzian metric g = dt2 − dx2 , where t ∈ R, x ∈ S 1 . This will simultaneously serve as a brief introduction to the idea of quantizing gauge theories after symplectic reduction, which will also be important in 3d quantum gravity. This approach is an alternative to the path-integral approach of the previous section, and in some cases is easier to make rigorous. 10
Any G-bundle P → R × S 1 is trivial, so we fix a trivialization and identify a connection on P with a g-valued 1-form on R × S 1 . The classical configuration space of the theory is the space A of connections on P |{0}×S 1 . This may be identified with the space of g-valued 1-forms on S 1 . The classical phase space of the theory is the cotangent bundle T ∗ A. Note that a tangent vector v ∈ TA A may be identified with a g-valued 1-forms on S 1 . We may also think of a g-valued 1-form E on S 1 as a cotangent vector, using the nondegenerate inner product: hE, vi = −
Z
S1
tr(E ∧ ⋆v),
We thus regard the phase space T ∗ A as the space of pairs (A, E) of g-valued 1-forms on S 1 . Given a connection on P solving the Yang-Mills equation we obtain a point (A, E) of the phase space T ∗ A as follows: let A be the pullback of the connection to {0}×S 1 , and let E be its covariant time derivative pulled back to {0} × S 1 . The pair (A, E) is called the initial data for the solution, and in physics A is called the vector potential and E the electric field. The Yang-Mills equation implies a constraint on (A, E), the Gauss law dA ⋆ E = 0, and any pair (A, E) satisfying this constraint is the initial data for some solution of the Yang-Mills equation. However, this solution is not unique, due to the gaugeinvariance of the equation. Moreover, the loop group G = C ∞ (S 1 , G) acts as gauge transformations on A, and this action lifts naturally to an action on T ∗ A, given by: g: (A, E) → (gAg −1 + gd(g −1), gEg −1). Two points in the phase space T ∗ A are to be regarded as physically equivalent if they differ by a gauge transformation. In this sort of situation it is natural to try to construct a smaller, more physically relevant “reduced phase space” using the process of symplectic reduction. The phase space T ∗ A is a symplectic manifold, but the constraint subspace {(A, E)| dA ⋆ E = 0} ⊂ T ∗ A is not. However, the constraint dA ⋆ E, integrated against any f ∈ C ∞ (S 1 , g) as follows, Z tr(f dA ⋆ E), S1
gives a function on phase space that generates a Hamiltonian flow coinciding with a one-parameter group of gauge transformations. In fact, all one-parameter subgroups of G are generated by the constraint in this fashion. Consequently, the quotient of the constraint subspace by G is again a symplectic manifold, the reduced phase space.
11
In the case at hand there is a very concrete description of the reduced phase space. First, by basic results on moduli spaces of flat connections, the “reduced configuration space” A/G may be naturally identified with Hom(π1 (S 1 ), G)/AdG, which is just G/AdG. Alternatively, one can see this quite concretely. We may first take the quotient of A by only those gauge transformations that equal the identity at a given point of S 1 : G0 = {g ∈ C ∞ (S 1 , G)| g(0) = 1}. This “almost reduced” configuration space A/G0 is diffeomorphic to G itself, with an explicit diffeomorphism taking each equivalence class [A] to its holonomy around the circle: H [A] 7→ P e S 1 A The remaining gauge transformations form the group G/G0 ∼ = G, which acts on the almost reduced configuration space G by conjugation, so A/G ∼ = G/AdG. ∞ Next, writing E = edx, the Gauss law says that e ∈ C (S 1 , g) is a flat section, hence determined by its value at the basepoint of S 1 . It follows that any point (A, E) in the constraint subspace is determined by A ∈ A together with e(0) ∈ g. The quotient of the constraint subspace by G0 , the “almost reduced” phase space, is thus identified with T ∗ G. It follows that the quotient of the constraint subspace by all of G, the reduced phase space, is identified with T ∗ G/AdG. The advantage of the almost reduced configuration space and phase space is that they are manifolds. Observables of the classical theory can be identified either with functions on the reduced phase space, or functions on the almost reduced phase space T ∗ G that are constant on the orbits of the lift of the adjoint action of G. For example, the Yang-Mills Hamiltonian is initially a function on T ∗ A: 1 H(A, E) = hE, Ei 2 but by the process of symplectic reduction one obtains a corresponding Hamiltonian on the reduced phase space. One can, however, carry out only part of the process of symplectic reduction, and obtain a Hamiltonian function on the almost reduced phase space. This is just the Hamiltonian for a free particle on G, i.e., for any p ∈ Tg∗ G it is given by 1 H(g, p) = kpk2 2 ∗ with the obvious inner product on Tg G. Now let us consider quantizing 2-dimensional Yang-Mills theory. What should be the Hilbert space for the quantized theory on R × S 1 ? As described in the previous section, it is natural to take L2 of the reduced configuration space A/G. (Since the theory is not generally covariant, the diffeomorphism and Hamiltonian constraints do not enter; the “kinematical” Hilbert space is the physical Hilbert space.) However, to 12
define L2 (A/G) requires choosing a measure on A/G = G/AdG. We will choose the pushforward of normalized Haar measure on G by the quotient map G → G/AdG. This measure has the advantage of mathematical elegance. While one could also argue for it on physical grounds, we prefer to simply show ex post facto that it gives an interesting quantum theory consistent with other approaches to 2d Yang-Mills theory. To begin with, note that this measure gives a Hilbert space isomorphism L2 (A/G) ∼ = L2 (G)inv where the right side denotes the subspace of L2 (G) consisting of functions constant on each conjugacy class of G. Let χρ denote the character of an equivalence class ρ of irreducible representations of G. Then by the Schur orthogonality relations, the set {χρ } forms an orthonormal basis of L2 (G)inv . In fact, the Hamiltonian of the quantum theory is diagonalized by this basis. Since the Yang-Mills Hamiltonian Hamiltonian on the almost reduced phase space T ∗ G is that of a classical free particle on G, we take the quantum Hamiltonian to be that for a quantum free particle on G: H = ∆/2 where ∆ is the (nonnegative) Laplacian on G. When we decompose the regular representation of G into irreducibles, the function χρ lies in the sum of copies of the representation ρ, so 1 Hχρ = c2 (ρ)χρ , 2
(1)
where c2 (ρ) is the quadratic Casimir of G in the representation ρ. Note that the vacuum (the eigenvector of H with lowest eigenvalue) is the function 1, which is χρ for ρ the trivial representation. In a sense this diagonalization of the Hamiltonian completes the solution of YangMills theory on R × S 1 . However, extracting the physics from this solution requires computing expectation values of physically interesting observables. To take a step in this direction, and to make the connection to string theory, let us consider the Wilson loop observables. Recall that given a based loop γ: S 1 → S 1 , the classical Wilson loop T (γ, A) is defined by H T (γ, A) = trPe
γ
A
.
We may think of T (γ) = T (γ, ·) as a function on the reduced configuration space A/G, but it lifts to a function on the almost reduced configuration space G, and we prefer to think of it as such. In the case at hand these Wilson loop observables depend only on the homotopy class of the loop, because all connections on S 1 are flat. In the string field picture of Section 2, we obtain a theory in which all physical states have ψ(η1 , · · · , ηn ) = ψ(γ1 , · · · , γn ) 13
when ηi is homotopic to γi for all i. We will see this again in 3d quantum gravity. Letting γn : S 1 → S 1 be an arbitrary loop of winding number n, we have T (γn , g) = tr(g n ). Since the classical Wilson loop observables are functions on configuration space, we may quantize them by interpreting them as multiplication operators acting on L2 (G)inv : (T (γn )ψ)(g) = tr(g n )ψ(g). We can also form elements of L2 (G)inv by applying products of these operators to the vacuum. Let |n1 , . . . , nk i = T (γn1 ) · · · T (γnk )1 The states |n1 , . . . , nk i may also be regarded as states of a string theory in which k strings are present, with winding numbers n1 , . . . , nk , respectively. For convenience, we define |∅i to be the vacuum state. The resemblance of the “string states” |n1 , . . . , nk i to states in a bosonic Fock space should be clear. In particular, the T (γn ) are analogous to “creation operators.” However, we do not generally have a representation of the canonical commutation relations. In fact, the string states do not necessarily span L2 (G)inv , although they do in some interesting cases. They are never linearly independent, because the Wilson loops satisfy relations. One always has T (γ0) = tr(1), for example, and for any particular group G the Wilson loops will satisfy identities called Mandelstam identities. For example, for G = SU(2) and taking traces in the fundamental representation, the Mandelstam identity is T (γn )T (γm) = T (γn+m ) + T (γn−m ). Note that this implies that |n, mi = |n + mi + |n − mi, so the total number of strings present in a given state is ambiguous. In other words, there is no analog of the Fock space “number operator” on L2 (G)inv . String states appear prominently in the work of Gross, Taylor, Minahan and Polychronakos [25, 35] on SU(N) Yang-Mills theory in 2 dimensions as a string theory. These authors, however, work primarily with the large N limit of SU(N) Yang-Mills theory, for since the work of t’Hooft [47] it has been clear that SU(N) Yang-Mills theory simplifies as N → ∞. In what follows we will use many ideas from these authors, but give a string-theoretic formula for the SU(N) Yang-Mills Hamiltonian that is exact for arbitrary N, instead of working in the large N limit. For the rest of this section we set G = SU(N) and take traces in the fundamental representation. In this case the string states do span L2 (G)inv , and all the linear
14
dependencies between string states are consequences of the following Mandelstam identities [20]. Given loops η1 , . . . , ηk in S 1 , let Mk (η1 , . . . , ηk ) =
1 X sgn(σ)T (gj11 · · · gj1n1 ) · · · T (gjk1 · · · gjknk ) k! σ∈Sk
where σ has the cycle structure (j11 · · · j1n1 ) · · · (jk1 · · · jknk ). Then MN (η, . . . , η) = 1 for all loops η, and MN +1 (η1 , . . . , ηN +1 ) = 0 for all loops ηi . There are also explicit formulas expressing the string states in terms of the basis {χρ } of characters. These formulas are based on the classical theory of Young diagrams, which we shall briefly review. The importance of this theory for 2d Yang-Mills theory is clear from the work of Gross and Taylor [24, 25]. As we shall see, Young diagrams describe a “duality” between the representation theory of SU(N) and of the symmetric groups Sn which can be viewed as a mathematical reflection of string field/gauge field duality. First, note using the Mandelstam identities that the string states |n1 , · · · , nk i with all the ni positive (but k possibly equal to zero) span L2 (SU(N))inv . Thus we will restrict our attention for now to states of this kind, which we call “right-handed.” There is a 1-1 correspondence between right-handed string states and conjugacy classes of permutations in symmetric groups, in which the string state |n1 , · · · , nk i corresponds to the conjugacy class σ of all permutations with cycles of length n1 , · · · , nk . Note that σ consists of permutations in Sn(σ) , where n(σ) = n1 +· · ·+nk . To take advantage of this correspondence, we simply define |σi = |n1 , · · · , nk i. when σ is the conjugacy class of permutations with cycle lengths n1 , . . . , nk . We will assume without loss of generality that n1 ≥ · · · ≥ nk > 0. The rationale for this description of string states as conjugacy classes of permutations is in fact quite simple. Suppose we have length-minimizing strings in S 1 with winding numbers n1 , . . . , nk . Labelling each strand of string each time it crosses the point x = 0, for a total of n = n1 + · · · + nk labels, and following the strands around counterclockwise to x = 2π, we obtain a permutation of the labels, hence an element of Sn . However, since the labelling was arbitrary, the string state really only defines a conjugacy class σ of elements of Sn . In a Young diagram one draws a conjugacy class σ with cycles of length n1 ≥ · · · ≥ nk > 0 as a diagram with k rows of boxes, having ni boxes in the ith row. (See Figure 2.) Let Y denote the set of Young diagrams.
15
Figure 2: Young Diagram On the one hand, there is a map from Young diagrams to equivalence classes of irreducible representations of SU(N). Given ρ ∈ Y , we form an irreducible representation of SU(N), which we also call ρ, by taking a tensor product of n copies of the fundamental representation, one copy for each box, and then antisymmetrizing over all copies in each column and symmetrizing over all copies in each row. This gives a 1-1 correspondence between Young diagrams with < N rows and irreducible representations of SU(N). If ρ has N rows it is equivalent to a representation coming from a Young diagram having < N rows, and if ρ has > N rows it is zero-dimensional. We will write χρ for the character of the representation ρ; if ρ has > N rows χρ = 0. On the other hand, Young diagrams with n boxes are in 1-1 correspondence with irreducible representations of Sn . This allows us to write the Frobenius relations expressing the string states |σi in terms of characters χρ and vice versa. Given ρ ∈ Y , we write ρ˜ for the corresponding representation of Sn . We define the function χρ˜ on Sn to be zero for n(ρ) 6= n, where n(ρ) is the number of boxes in ρ. Then the Frobenius relations are |σi =
X
χρ˜(σ)χρ ,
(2)
ρ∈Y
and conversely χρ =
X 1 χρ˜(σ) |σi. n(ρ)! σ∈Sn(ρ)
(3)
The Yang-Mills Hamiltonian has a fairly simple description in terms of the basis of characters {χρ }. First, recall that equation (1) expresses the Hamiltonian in terms 16
of the Casimir. There is an explicit formula for the value of the SU(N) Casimir in the representation ρ: c2 (ρ) = Nn(ρ) − N −1 n(ρ)2 +
n(ρ)(n(ρ) − 1)χρ˜(“2”) dim(˜ ρ)
where “2” denotes the conjugacy class of permutations in Sn(ρ) with one cycle of length 2 and the rest of length 1. It follows that 1 H = (NH0 − N −1 H02 + H1 ) 2
(4)
H0 χρ = n(ρ)χρ
(5)
n(ρ)(n(ρ) − 1)χρ (“2”) χρ . dim(˜ ρ)
(6)
where
and H1 χρ =
To express the operators H0 and H1 in string-theoretic terms, it is convenient to define string annihilation and creation operators satisfying the canonical commutation relations. As noted above, there is no natural way to do this in L2 (SU(N))inv since the string states are not linearly independent. The work of Gross, Polychronakos relies on the fact that any finite set of distinct string states becomes linearly independent, in fact orthogonal, for sufficiently large N. We will proceed slightly differently, simply defining a space in which all the string states are independent. Let H be a Hilbert space having an orthonormal basis {Xρ }ρ∈Y indexed by all Young diagrams. For each σ ∈ Y , define a vector |σi in H by the Frobenius relation (2). Then a calculation using the Schur orthogonality equations twice shows that these string states |σi are, not only linearly independent, but orthogonal: hσ|σ ′ i =
X
χρ˜(σ)χρ˜′ (σ ′ )hXρ , Xρ′ i
ρ,ρ′ ∈Y
=
X
χρ˜(σ)χρ˜(σ ′ )
ρ∈Y
=
n(σ)! δσσ′ . |σ|
where |σ| is the number of elements in σ regarded as a conjugacy class in Sn . One can also derive the Frobenius relation (3) from these definitions and express the basis {Xρ } in terms of the string states: Xρ =
X 1 χρ˜(σ)|σi. n(ρ)! σ∈Sn(ρ)
17
It follows that the string states form a basis for H. The Yang-Mills Hilbert space L2 (SU(N))inv is a quotient space of the string field Hilbert space H, with the quotient map j: H → L2 (SU(N))inv being given by Xρ 7→ χρ . This quotient map sends the string state |σi in H to the corresponding string state |σi ∈ L2 (SU(N))inv . It follows that this quotient map is precisely that which identifies any two string states that are related by the Mandelstam identities. It was noted some time ago by Gliozzi and Virasoro [22] that Mandelstam identities on string states are strong evidence for a gauge field interpretation of a string field theory. Here in fact we will show that the Hamiltonian on the Yang-Mills Hilbert space L2 (SU(N))inv lifts to a Hamiltonian on H with a simple interpretation in terms of string interactions, so that 2-dimensional SU(N) Yang-Mills theory is isomorphic to a quotient of a string theory by the Mandelstam identities. In the framework of the previous section, the Mandelstam identities would appear as part of the “dynamical constraint” K of the string theory. Following equations (4-6), we define a Hamiltonian H on the string field Hilbert space H by 1 H = (NH0 − N −1 H02 + H1 ) 2 where n(n − 1)χρ˜(“2”) Xρ . H0 Xρ = n(ρ)Xρ , H1 Xρ = dim(˜ ρ) This clearly has the property that Hj = jH, so the Yang-Mills dynamics is the quotient of the string field dynamics. On H we can introduce creation operators a∗j (j > 0) by a∗j |n1 , . . . , nk i = |j, n1 , · · · , nk i, and define the annihilation operator aj to be the adjoint of a∗j . These satisfy the following commutation relations: [aj , ak ] = [a∗j , a∗k ] = 0,
[aj , a∗k ] = jδjk .
We could eliminate the factor of j and obtain the usual canonical commutation relations by a simple rescaling, but it is more convenient not to. We then claim that H0 =
X
j>0
18
a∗j aj
H1 ψ( @ R ) = ψ( @
-
)
Figure 3: Two-string interaction in 1-dimensional space and H1 =
X
a∗j+k aj ak + a∗j a∗k aj+k .
j,k>0
These follow from calculations by Minahan and Polychronakos [35], which we briefly sketch here. The Frobenius relations and the definition of H0 give H0 |σi = n(σ)|σi,
(7)
and this implies the formula for H0 as a sum of harmonic oscillator Hamiltonians a∗j aj . Similarly, the Frobenius relations and the definition of H1 give H1 |σi =
X
ρ∈Y
n(σ)(n(σ) − 1) χρ˜(“2”)χρ˜(σ)χρ . dim(˜ ρ)
Since there are n(n − 1)/2 permutations τ ∈ Sn(σ) lying in the conjugacy class “2”, we may rewrite this as H1 |σi = Since
X
2 χρ˜(σ)χρ˜(τ )χρ . ρ) ρ∈Y,τ ∈“2” dim(˜
X 1 χρ˜(σ)χρ˜(τ ) = χρ˜(στ ) ρ) τ ∈“2” τ ∈“2” dim(˜ X
the Frobenius relations give H1 |σi = 2
X
|στ i.
(8)
τ ∈“2”
An analysis of the effect of composing σ with all possible τ ∈ “2” shows that either one cycle of σ will be broken into two cycles, or two will be joined to form one, giving the expression above for H1 in terms of annihilation and creation operators. We may interpret the Hamiltonian in terms of strings as follows. By equation (7), H0 can be regarded as a “string tension” term, since if we represent a string state |n1 , . . . , nk i by length-minimizing loops, it is an eigenvector of H0 with eigenvalue equal to n1 + · · · + nk , proportional to the sum of the lengths of the loops. By equation (8), H1 corresponds to a two-string interaction as in Figure 3. In this figure only the x coordinate is to be taken seriously; the other has been introduced only to keep track of the identities of the strings. Also, we have switched to treating states as functions on the space of multiloops. As the figure indicates, this kind of 19
interaction is a 1-dimensional version of that which gave the HOMFLY invariant of links in 3-dimensional space in the previous section. Here, however, we have a true Hamiltonian rather than a Hamiltonian constraint. Figure 3 can also be regarded as two frames of a “movie” of a string worldsheet in 2dimensional spacetime. Similar movies have been used by Carter and Saito to describe string worldsheets in 4-dimensional spacetime [13]. If we draw the string worldsheet corresponding to this movie we obtain a surface with a branch point. Indeed, in the path integral approach of Gross and Taylor this kind of term appears in the partition function as part of a sum over string histories, associated to those histories with branch points. They also show that the H02 term corresponds to string worldsheets with handles. When considering the 1/N expansion of the theory, it is convenient to divide the Hamiltonian H by N, so that it converges to H0 as N → ∞. Then the H02 term is proportional to 1/N 2 . This is in accord with the observation by t’Hooft [47] that in an expansion of the free energy (logarithm of the partition function) as a power series in 1/N, string worldsheets of genus g give terms proportional to 1/N 2−2g . From the work of Gross and Taylor it is also clear that in addition to the space H spanned by right-handed string states one should also consider a space with a basis of “left-handed” string states |n1 , · · · , nk i with ni < 0. The total Hilbert space of the string theory is then the tensor product H+ ⊗ H− of right-handed and lefthanded state spaces. This does not describe any new states in the Yang-Mills theory per se, but it is more natural from the string-theoretic point of view. It follows from the work of Minahan and Polychronakos that there is a Hamiltonian H on H+ ⊗ H− naturally described in terms of string interactions and a quotient map j: H+ ⊗ H− → L2 (SU(N))inv such that Hj = jH.
4
Quantum Gravity in 3 dimensions
Now let us turn to a more sophisticated model, 3-dimensional quantum gravity. In 3 dimensions, Einstein’s equations say simply that the spacetime metric is flat, so there are no local degrees of freedom. The theory is therefore only interesting on topologically nontrivial spacetimes. Interest in the mathematics of this theory increased when Witten [49] reformulated it as a Chern-Simons theory. Since then, many approaches to the subject have been developed, not all equivalent [11]. We will follow Ashtekar, Husain, Rovelli, Samuel and Smolin [1, 6] and treat 3-dimensional quantum gravity using the “new variables” and the loop transform, and indicate some possible relations to string theory. It is important to note that there are some technical problems with the loop transform in Lorentzian quantum gravity, since the gauge group is then noncompact [32]. These are presently being addressed by Ashtekar and Loll [5] in the 3-dimensional case, but for simplicity of presentation we will sidestep them by working with the Riemannian case, where the gauge group is SO(3). It is easiest to describe the various action principles for gravity using the abstract index notation popular in general relativity, but we will instead translate them into 20
language that may be more familiar to mathematicians, since this seems not to have been done yet. In this section we describe the “Witten action,” applicable to the 3dimensional case; in the next section we describe the “Palatini action,” which applies to any dimension, and the “Ashtekar action,” which applies to 4 dimensions. The relationship between these action principles has been discussed rather thoroughly by Peldan [39]. Let the spacetime M be an orientable 3-manifold. Fix a real vector bundle T over M that is isomorphic to - but not canonically identified with - the tangent bundle T M, and fix a Riemannian metric η and an orientation on T . These define a “volume form” ǫ on T , that is, a nowhere vanishing section of Λ3 T ∗ . The basic fields of the theory are then taken to be a metric-preserving connection A on T , or “SO(3) connection,” together with a T -valued 1-form e on M. Using the isomorphism T ∼ = T ∗ given by 2 the metric, the curvature F of A may be identified with a Λ T -valued 2-form. It follows that the wedge product e ∧ F may may be defined as a Λ3 T -valued 3-form. Pairing this with ǫ to obtain an ordinary 3-form and then integrating over spacetime, we obtain the Witten action 1 S(A, e) = 2
Z
M
ǫ(e ∧ F ).
The classical equations of motion obtained by extremizing this action are F =0 and dA e = 0. Note that we can pull back the metric η on E by e: T M → T to obtain a “Riemannian metric” on M, which, however, is only nondegenerate when e is an isomorphism. When e is an isomorphism we can also use it to pull back the connection to a metricpreserving connection on T M. In this case, the equations of motion say simply that this connection is the Levi-Civita connection of the metric on M, and that the metric on M is flat. The formalism involving the fields A and e can thus be regarded as a device for extending the usual Einstein equations in 3 dimensions to the case of degenerate “metrics” on M. Now suppose that M = R × X, where X is a compact oriented 2-manifold. The classical configuration and phase spaces and their reduction by gauge transformations are reminiscent of those for 2d Yang-Mills theory. There are, however, a number of subtleties, and we only present the final results. The classical configuration space can be taken as the space A of metric-preserving connections on T |X, which we call SO(3) connections on X. The classical phase space is then the cotangent bundle T ∗ A. Note that a tangent vector v ∈ TA A is a Λ2 T -valued 1-form on X. We can thus regard a T -valued 1-form E˜ on X as a cotangent vector by means of the pairing ˜ E(v) =
Z
X
ǫ(E˜ ∧ v).
21
Thus given any solution (A, e) of the classical equations of motion, we can pull back A and e to the surface {0} × X and get an SO(3) connection and a T -valued 1-form ˜ where on X, that is, a point in the phase space T ∗ A. This is usually written (A, E), E˜ plays a role analogous to the electric field in Yang-Mills theory. ˜ ∈ T ∗ A which define The classical equations of motion imply constraints on (A, E) a reduced phase space. These are the Gauss law, which in this context is dA E˜ = 0, and the vanishing of the curvature B of the connection A on T |X, which is analogous to the magnetic field: B = 0. The latter constraint subsumes both the diffeomorphism and Hamiltonian constraints of the theory. The reduced phase space for the theory turns out to be T ∗ (A0 /G), where A0 is the space of flat SO(3) connections on X, and G is the group of gauge transformations [6]. As in 2d Yang-Mills theory, it will be attractive quantize after imposing constraints, taking the physical state space of the quantized theory to be L2 of the reduced configuration space, if we can find a tractable description of A0 /G. A quite concrete description of A0 /G was given by Goldman [23]. The moduli space F of flat SO(3)-bundles has two connected components, corresponding to the two isomorphism classes of SO(3) bundles on M. The component corresponding to the bundle T |X is precisely the space A0 /G, so we wish to describe this component. There is a natural identification F∼ = Hom(π1 (X), SO(3))/Ad(SO(3)), given by associating to any flat bundle the holonomies around (homotopy classes of) loops. Suppose that X has genus g. Then the group π1 (X) has a presentation with 2g generators x1 , y1 , . . . , xg , yg satisfying the relation −1 −1 −1 R(xi , yi ) = (x1 y1 x−1 1 y1 ) · · · (xg yg xg yg ) = 1.
An element of Hom(π1 (X), SO(3)) may thus be identified with a collection u1 , v1 , . . . , ug , vg of elements of SO(3), satisfying R(ui, vi ) = 1, and a point in F is an equivalence class [ui, vi ] of such collections. The two isomorphism classes of SO(3) bundles on M are distinguished by their second Stiefel-Whitney number w2 ∈ Z2 . The bundle T |X is trivial so w2 (T |X) = 0 We can calculate w2 for any point [ui , vi ] ∈ F by the following method. For all the g ∼ elements ui , vi ∈ SO(3), choose lifts u˜i , v˜i to the universal cover SO(3) = SU(2). Then (−1)w2 = R(˜ ui , v˜i ). 22
It follows that we may think of points of A0 /G as equivalence classes of 2g-tuples (ui , vi ) of elements of SO(3) admitting lifts u˜i , v˜i with R(˜ ui, v˜i ) = 1, where the equivalence relation is given by the adjoint action of SO(3). In fact A0 /G is, not a manifold, but a singular variety. This has been investigated by Narasimhan and Seshadri [38], and shown to be dimension d = 6g − 6 for g ≥ 2, or d = 2 for g = 1 (the case g = 0 is trivial and will be excluded below). As noted, it is natural to take L2 (A0 /G) to be the physical state space, but but to define this one must choose a measure on A0 /G. As noted by Goldman [23], there is a symplectic structure Ω on A0 /G coming from the following 2-form on A0 : Ω(B, C) =
Z
X
tr(B ∧ C),
in which we identify the tangent vectors B, C with End(T |X)-valued 1-forms. The d-fold wedge product Ω ∧ · · · ∧ Ω defines a measure µ on A0 /G, the Liouville measure. On the grounds of elegance and diffeomorphism-invariance it is customary to use this measure to define the physical state space L2 (A0 /G). It would be satisfying if there were a string-theoretic interpretation of the inner product in L2 (A0 /G) along the lines of Section 2. Note that we may define “string states” in this space as follows. Given any loop γ in X, the Wilson loop observable T (γ) is a multiplication operator on L2 (A0 /G) that only depends on the homotopy class of γ. As in the case of 2d Yang-Mills theory, we can form elements of L2 (A0 /G) by applying products of these operators to the function 1, so given γ1 , · · · , γk ∈ π1 (X), define |γ1 , . . . , γk i = T (γ1 ) · · · T (γk )1 The first step towards a string-theoretic interpretation of 3d quantum gravity would be a formula for inner products of the form hγ1 , . . . , γk |γ1′ , . . . , γk′ ′ i, or, equivalently, for integrals of the form Z
A0 /G
T (γ1 , A) · · · T (γk , A)dµ(A).
The author has been unable to find such a formula in the literature except for the case g = 1. Note that this sort of integral makes sense taking A0 to be the space of flat connections for a trivial SO(N) bundle over X, for any N. Alternatively, one could formulate 3d quantum gravity as a theory of SU(2) connections and then generalize to SU(N). One might expect that, as in 2d Yang-Mills theory, the situation simplifies in the N → ∞ limit. Ideally, one would like a formula for hγ1 , . . . , γk |γ1′ , . . . , γk′ ′ i 23
in the N → ∞ limit, together with a method of treating the finite N case by imposing Mandelstam identities. In the N → ∞ limit one would also hope for a formula in terms of a sum over ambient isotopy classes of surfaces f : Σ → [0, T ] × X having the loops γi, γi′ as boundaries. Before concluding this section, it is worth noting another generally covariant gauge theory in 3 dimensions, Chern-Simons theory. Here one fixes an arbitrary Lie group G and a G-bundle P → M over spacetime, and the field of the theory is a connection A on P . The action is given by k S(A) = 4π
Z
2 tr(A ∧ dA + A ∧ A ∧ A). 3
As noted by Witten [49], 3d quantum gravity as we have described it is essentially the same Chern-Simons theory with gauge group ISO(3), the Euclidean group in 3 dimension, with the SO(3) connection and triad field appearing as two parts of an ISO(3) connection. There is a profound connection between Chern-Simons theory and knot theory, first demonstrated by Witten [48], and then elaborated by many researchers (see, for example, [7]). This theory does not quite fit our formalism because in it the space A0 /G of flat connections modulo gauge transformations plays the role of a phase space, with the Goldman symplectic structure, rather than a configuration space. Nonetheless, there are a number of clues that Chern-Simons theory admits a reformulation as a generally covariant string field theory. In fact, Witten has given such an interpretation using open strings and the Batalin-Vilkovisky formalism [50]. Moreover, for the gauge groups SU(N) Periwal has expressed the partition function for Chern-Simons theory on S 3 , in the N → ∞ limit, in terms of integrals over moduli spaces of Riemann surfaces. In the case N = 2 there is also, as one would expect, an expression for the vacuum expectation value of Wilson loops, at least for the case of a link (where it is just the Kauffman bracket invariant), in terms of a sum over surfaces having that link as boundary [12]. It would be very worthwhile to reformulate Chern-Simons theory as a string theory at the level of elegance with which one can do so for 2d Yang-Mills theory, but this has not yet been done.
5
Quantum Gravity in 4 dimensions
We begin by describing the Palatini and Ashtekar actions for general relativity. As in the previous section, we will sidestep certain problems with the loop transform by working with Riemannian rather than Lorentzian gravity. We shall then discuss some recent work on making the loop representation rigorous in this case, and indicate some mathematical issues that need to be explored to arrive at a string-theoretic interpretation of the theory. Let the spacetime M be an orientable n-manifold. Fix a bundle T over M that is isomorphic to T M, and fix a Riemannian metric η and orientation on T . These define a nowhere vanishing section ǫ of Λn T ∗ . The basic fields of the theory are then 24
taken to be a metric-preserving connection A on T , or “SO(n) connection,” and a T -valued 1-form e. We require, however, that e: T M → T be a bundle isomorphism; its inverse is called a “frame field.” The metric η defines to an isomorphism T ∼ =T∗ and allows us to identify the curvature F of A with a section of the bundle Λ2 T ⊗ Λ2 T ∗ M. We may also regard e−1 as a section of T ∗ ⊗ T M and define e−1 ∧ e−1 in the obvious manner as a section of the bundle Λ2 T ∗ ⊗ Λ2 T M. The natural pairing between these bundles gives rise to a function F (e−1 ∧ e−1 ) on M. Using the isomorphism e, we can push forward ǫ to a volume form ω on M. The Palatini action for Riemannian gravity is then 1Z S(A, e) = F (e−1 ∧ e−1 ) ω. 2 M We may use the isomorphism e to transfer the metric η and connection A to a metric and connection on the tangent bundle. Then the classical equations of motion derived from the Palatini action say precisely that this connection is the Levi-Civita connection of the metric, and that the metric satisfies the vacuum Einstein equations (i.e., is Ricci flat). In 3 dimensions, the Palatini action reduces to the Witten action, which however is expressed in terms of e rather than e−1 . In 4 dimensions the Palatini action can be rewritten in a somewhat similar form. Namely, the wedge product e ∧ e ∧ F is a Λ4 T -valued 4-form, and pairing it with ǫ to obtain an ordinary 4-form we have 1 S(A, e) = 2
Z
M
ǫ(e ∧ e ∧ F ).
The Ashtekar action depends upon the fact that in 4 dimensions the metric and orientation on T define a Hodge star operator ∗: Λ2 T → Λ2 T with ∗2 = 1 (not to be confused with the Hodge star operator on differential forms). This allows us to write F as a sum F+ + F− of self-dual and anti-self-dual parts: ∗F± = ±F± . The remarkable fact is that the action S(A, e) =
1 2
Z
M
25
ǫ(e ∧ e ∧ F+ )
gives the same equations of motion as the Palatini action. Moreover, suppose T is trivial, as is automatically the case when M ∼ = R × X. Then F is just an so(4)valued 2-form on M, and its decomposition into self-dual and anti-self-dual parts corresponds to the decomposition so(4) ∼ = so(3) ⊕ so(3). Similarly, A is an so(4)valued 1-form, and may thus be written as a sum A+ + A− of “self-dual” and “antiself-dual” connections, which are 1-forms having values in the two copies of so(3). It is easy to see that F+ is the curvature of A+ . This allows us to regard general relativity as the theory of a self-dual connection A+ and a T -valued 1-form e - the so-called “new variables” - with the Ashtekar action S(A+ , e) =
1 2
Z
M
ǫ(e ∧ e ∧ F+ ).
Now suppose that M = R × X, where X is a compact oriented 3-manifold. We can take the classical configuration space to be space A of right-handed connections on T |X, or equivalently (fixing a trivialization of T ), so(3)-valued 1-forms on X. A tangent vector v ∈ TA A is thus an so(3)-valued 1-form, and an so(3)-valued 2-form E˜ defines a cotangent vector by the pairing ˜ E(v) =
Z
X
˜ ∧ v). tr(E
˜ consisting of an so(3)A point in the classical phase space T ∗ A is thus a pair (A, E) ˜ on X. In the physics literature it is valued 1-form A and an so(3)-valued 2-form E more common to use the natural isomorphism Λ2 T ∗ X ∼ = T X ⊗ Λ3 T ∗ X ˜ as an given by the interior product to regard the “gravitational electric field” E 3 ∗ so(3)-valued vector density, that is, a section of so(3) ⊗ T X ⊗ Λ T X. ˜ ∈ A solution (A+ , e) of the classical equations of motion determines a point (A, E) T ∗ A as follows. The “gravitational vector potential” A is simply the pullback of A+ to the surface {0} × X. Obtaining E˜ from e is a somewhat subtler affair. First, split the bundle T as the direct sum of a 3-dimensional bundle 3 T and a line bundle. By restricting to T X and then projecting down to 3 T |X, the map e: T M → T gives a map 3
e: T X → 3 T |X,
called a “cotriad field” on X. Since there is a natural isomorphism of the fibers of 3 T with so(3), we may also regard this as an so(3)-valued 1-form on X. Applying the ˜ Hodge star operator we obtain the so(3)-valued 2-form E. ˜ ∈ T ∗ A. These are The classical equations of motion imply constraints on (A, E) the Gauss law dA E˜ = 0, 26
and the diffeomorphism and Hamiltonian constraints. The latter two are most easily expressed if we treat E˜ as an so(3)-valued vector density. Letting B denote the “gravitational magnetic field,” or curvature of the connection A, the diffeomorphism constraint is given by tr iE˜ B = 0 and the Hamiltonian constraint is given by tr iE˜ iE˜ B = 0. Here the interior product iE˜ B is defined using 3 × 3 matrix multiplication and is a M3 (R)⊗Λ3 T ∗ X-valued 1-form; similarly, iE˜ iE˜ B is a M3 (R)⊗Λ3 T ∗ X ⊗Λ3 T ∗ X-valued function. In 2d Yang-Mills theory and 3d quantum gravity one can impose enough constraints before quantizing to obtain a finite-dimensional reduced configuration space, namely the space A0 /G of flat connections modulo gauge transformations. In 4d quantum gravity this is no longer the case, so a more sophisticated strategy, first devised by Rovelli and Smolin [44], is required. Let us first sketch this without mentioning the formidable technical problems. The Gauss law constraint generates gauge transformations so one forms the reduced phase space T ∗ (A/G). Quantizing, one obtains the kinematical Hilbert space Hkin = L2 (A/G). One then applies the loop transform and takes Hkin = Fun(M) to be a space of functions of multiloops in X. The diffeomorphism constraint generates the action of Diff 0 (X) on A/G, so in the quantum theory one takes Hdif f to be the subspace of Diff 0 (X)-invariant elements of Fun(M). One may then either attempt to represent the Hamiltonian constraint as operators on Hkin , and define the image of their common kernel in Hdif f to be the physical state space Hphys , or attempt to represent the Hamiltonian constraint directly as operators on Hdif f and define the kernel to be Hphys . (The latter approach is still under development by Rovelli and Smolin [45].) Even at this formal level, the full space Hphys has not yet been determined. In their original work, Rovelli and Smolin [44] obtained a large set of physical states corresponding to ambient isotopy classes of links in X. More recently, physical states have been constructed from familiar link such as the Kauffman bracket and certain coefficients of the Alexander polynomial to all of M. Some recent developments along these lines have been reviewed by Pullin [42]. This approach makes use of the connection between 4d quantum gravity with cosmological constant and ChernSimons theory in 3 dimensions. It is this work that suggests a profound connection between knot theory and quantum gravity. There are, however, significant problems with turning all of this work into rigorous mathematics, so at this point we shall return to where we left off in Section 2 and discuss some of the difficulties. In Section 2 we were quite naive concerning many details of analysis - deliberately so, to indicate the basic ideas without becoming immersed in technicalities. In particular, one does not really expect to have interesting diffeomorphism-invariant measures on the space A/G of connections modulo 27
gauge transformations in this case. At best, one expects the existence of “generalized measures” sufficient for integrating a limited class of functions. In fact, it is possible to go a certain distance without becoming involved with these considerations. In particular, the loop transform can be rigorously defined without fixing a measure or generalized measure on A/G if one uses, not the Hilbert space formalism of the previous section, but a C*-algebraic formalism. A C*-algebra is an algebra A over the complex numbers with a norm and an adjoint or ∗ operation satisfying (a∗ )∗ = a,
(λa)∗ = λa∗ ,
(a + b)∗ = a∗ + b∗ ,
kabk ≤ kakkbk,
(ab)∗ = b∗ a∗ ,
ka∗ ak = kak2
for all a, b in the algebra and λ ∈ C. In the C*-algebraic approach to physics, observables are represented by self-adjoint elements of A, while states are elements µ of the dual A∗ that are positive, µ(a∗ a) ≥ 0, and normalized, µ(1) = 1. The number µ(a) then represents the expectation value of the observable a in the state µ. The relation to the more traditional Hilbert space approach to quantum physics is given by the Gelfand-Naimark-Segal (GNS) construction. Namely, a state µ on A defines an “inner product” that may however be degenerate: ha, bi = µ(a∗ b). Let I ⊆ A denote the subspace of norm-zero states. Then A/I has an honest inner product and we let H denote the Hilbert space completion of A/I in the corresponding norm. It is then easy to check that I is a left ideal of A, so that A acts by left multiplication on A/I, and that this action extends uniquely to a representation of A as bounded linear operators on H. In particular, observables in A give rise to self-adjoint operators on H. A C*-algebraic approach to the loop transform and generalized measures on A/G was introduced by Ashtekar and Isham [3] in the context of SU(2) gauge theory, and subsequently developed by Ashtekar, Lewandowski, and the author [4, 10]. The basic concept is that of the holonomy C*-algebra. Let X be a manifold, “space,” and let P → X be a principal G-bundle over X. Let A denote the space of smooth connections on P , and G the group of smooth gauge transformations. Fix a finitedimensional representation ρ of G and define Wilson loop functions T (γ) = T (γ, ·) on A/G taking traces in this representation. Define the “holonomy algebra” to be the algebra of functions on A/G generated by the functions T (γ) = T (γ, ·). If we assume that G is compact and ρ is unitary, the functions T (γ) are bounded and continuous (in the C ∞ topology on A/G). Moreover, the pointwise complex conjugate T (γ)∗ equals T (γ −1 ), where γ −1 is the orientationreversed loop. We may thus complete the holonomy algebra in the sup norm topology: kf k∞ = sup |f (A)| A∈A/G
28
and obtain a C*-algebra of bounded continuous functions on A/G, the “holonomy C*-algebra,” which we denote as Fun(A/G) in order to make clear the relation to the previous section. While in what follows we will assume that G is compact and ρ is unitary, it is important to emphasize that for Lorentzian quantum gravity G is not compact! This presents important problems in the loop representation of both 3- and 4-dimensional quantum gravity. Some progress in solving these problems has recently been made by Ashtekar, Lewandowski, and Loll [4, 5, 30]. Recall that in the previous section the loop transform of functions on A/G was defined using a measure on A/G. It turns out to be more natural to define the loop transform not on Fun(A/G) but on its dual, as this involves no arbitrary choices. Given µ ∈ Fun(A/G)∗ we define its loop transform µ ˆ to be the function on the space M of multiloops given by µ ˆ(γ1 , · · · , γn ) = µ(T (γ1) · · · T (γn )). Let Fun(M) denote the range of the loop transform. In favorable cases, such as G = SU(N) and ρ the fundamental representation, the loop transform is one-to-one, so Fun(A/G)∗ ∼ = Fun(M). This is the real justification for the term “string field/gauge field duality.” We may take the “generalized measures” on A/G to be simply elements µ ∈ Fun(A/G)∗ , thinking of the pairing µ(f ) as the integral of f ∈ Fun(A/G). If µ is a state on Fun(A/G), we may construct the kinematical Hilbert space Hkin using the GNS construction. Note that the kinematical inner product h[f ], [g]ikin = µ(f ∗ g) then generalizes the L2 inner product used in the previous section. Note that a choice of generalized measure µ also allows us to define the loop transform as a linear map from Fun(A/G) to Fun(M) fˆ(γ1 , · · · , γn ) = µ(T (γ1) · · · T (γn )f ) in a manner generalizing that of the previous section. Moreover, there is a unique inner product on Fun(M) such that this map extends to a map from Hkin to the Hilbert space completion of Fun(M). Note also that Diff 0 (X) acts on Fun(A/G) and dually on Fun(A/G)∗ . The kinematical Hilbert space constructed from a Diff 0 (X)invariant state µ ∈ Fun(A/G) thus becomes a unitary representation of Diff 0 (X). It is thus of considerable interest to find a more concrete description of Diff 0 (X)invariant states on the holonomy C*-algebra Fun(A/G). In fact, it is not immediately obvious that any exist, in general! For technical reasons, the most progress has been made in the real-analytic case. That is, we take X to be real-analytic, Diff 0 (X) to 29
consist of the real-analytic diffeomorphisms connected to the identity, and Fun(A/G) to be the holonomy C*-algebra generated by real-analytic loops. Here Ashtekar and Lewandowski have constructed a Diff 0 (X)-invariant state on Fun(A/G) that is closely analogous to the Haar measure on a compact group [4]. They have also given a general characterization of such diffeomorphism-invariant states. The latter was also given by the author [10], using a slightly different formalism, who also constructed many more examples of Diff 0 (X)-invariant states on Fun(A/G). There is thus some real hope that the loop representation of generally covariant gauge theories can be made rigorous in cases other than the toy models of the previous two sections. We conclude with some speculative remarks concerning 4d quantum gravity and 2-tangles. The correct inner product on the physical Hilbert space of 4d quantum gravity has long been quite elusive. A path-integral formula for the inner product has been investigated recently by Rovelli [43], but there is as yet no manifestly well-defined expression along these lines. On the other hand, an inner product for “relative states” of quantum gravity in the Kauffman bracket state has been rigorously constructed by the author [9], but there are still many questions about the physics here. The example of 2d Yang-Mills theory would suggest an expression for the inner product of string states hγ1, · · · , γn |γ1′ , · · · , γn′ i as a sum over ambient isotopy classes of surfaces f : Σ → [0, T ] × X having the loops γi , γi′ as boundaries. In the case of embeddings, such surfaces are known as “2-tangles,” and have been intensively investigated by Carter and Saito [13] using the technique of “movies.” The relationships between 2-tangles, string theory, and the loop representation of 4d quantum gravity are tantalizing but still rather obscure. For example, just as the Reideister moves relate any two pictures of the same tangle in 3 dimensions, there are a set of movie moves relating any two movies of the same 2-tangle in 4 dimensions. These moves give a set of equations whose solutions would give 2-tangle invariants. For example, the analog of the Yang-Baxter equation is the Zamolodchikov equation, first derived in the context of string theory [51]. These equations can be understood in terms of category theory, since just as tangles form a braided tensor category, 2tangles form a braided tensor 2-category [18]. It is thus quite significant that Crane [15] has initiated an approach to generally covariant field theory in 4 dimensions using braided tensor 2-categories. This approach also clarifies some of the significance of conformal field theory for 4-dimensional physics, since braided tensor 2-categories can be constructed from certain conformal field theories. In a related development, CottaRamusino and Martellini [14] have endeavored to construct 2-tangle invariants from generally covariant gauge theories, much as tangle invariants may be constructed using Chern-Simons theory. Clearly it will be some time before we are able to appraise the significance of all this work, and the depth of the relationship between string theory and the loop representation of quantum gravity.
30
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